LMFDB - Embedded newform 1305.2.d.b.811.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(811,1305)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1305.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.16516096.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 13x^{2} + 1 $ x^6 + 9x^4 + 13x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.1
Root			$-2.68667i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.b.811.6

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68667i q^{2} -5.21819 q^{4} -1.00000 q^{5} +4.21819 q^{7} +8.64620i q^{8} +O(q^{10})$	\(q-2.68667i q^{2} -5.21819 q^{4} -1.00000 q^{5} +4.21819 q^{7} +8.64620i q^{8} +2.68667i q^{10} +2.31446i q^{11} -0.643274 q^{13} -11.3329i q^{14} +12.7931 q^{16} -0.586195i q^{17} -5.95953i q^{19} +5.21819 q^{20} +6.21819 q^{22} +5.57491 q^{23} +1.00000 q^{25} +1.72826i q^{26} -22.0113 q^{28} +(5.21819 - 1.33061i) q^{29} -0.158221i q^{31} -17.0784i q^{32} -1.57491 q^{34} -4.21819 q^{35} -4.04272i q^{37} -16.0113 q^{38} -8.64620i q^{40} -9.76282i q^{41} -4.97568i q^{43} -12.0773i q^{44} -14.9779i q^{46} +2.31446i q^{47} +10.7931 q^{49} -2.68667i q^{50} +3.35673 q^{52} -13.0796 q^{53} -2.31446i q^{55} +36.4713i q^{56} +(-3.57491 - 14.0195i) q^{58} +2.64327 q^{59} +0.983848i q^{61} -0.425088 q^{62} -20.2978 q^{64} +0.643274 q^{65} +1.57491 q^{67} +3.05888i q^{68} +11.3329i q^{70} +9.35673 q^{71} +9.17663i q^{73} -10.8615 q^{74} +31.0980i q^{76} +9.76282i q^{77} -4.97568i q^{79} -12.7931 q^{80} -26.2295 q^{82} +8.21819 q^{83} +0.586195i q^{85} -13.3680 q^{86} -20.0113 q^{88} +7.29014i q^{89} -2.71345 q^{91} -29.0909 q^{92} +6.21819 q^{94} +5.95953i q^{95} -3.05888i q^{97} -28.9975i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 6 q^{5}+O(q^{10}) $ 6 * q - 6 * q^4 - 6 * q^5	$ 6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62} - 46 q^{64} + 4 q^{65} - 16 q^{67} + 56 q^{71} - 40 q^{74} - 26 q^{80} - 56 q^{82} + 24 q^{83} - 4 q^{86} - 44 q^{88} - 16 q^{91} - 48 q^{92} + 12 q^{94}+O(q^{100}) $ 6 * q - 6 * q^4 - 6 * q^5 - 4 * q^13 + 26 * q^16 + 6 * q^20 + 12 * q^22 + 8 * q^23 + 6 * q^25 - 56 * q^28 + 6 * q^29 + 16 * q^34 - 20 * q^38 + 14 * q^49 + 20 * q^52 - 28 * q^53 + 4 * q^58 + 16 * q^59 - 28 * q^62 - 46 * q^64 + 4 * q^65 - 16 * q^67 + 56 * q^71 - 40 * q^74 - 26 * q^80 - 56 * q^82 + 24 * q^83 - 4 * q^86 - 44 * q^88 - 16 * q^91 - 48 * q^92 + 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.68667i		−	1.89976i	−0.312613	−	0.949881i	$-0.601204\pi$
$2$		−	2.68667i		−	1.89976i	0.312613	−	0.949881i	$-0.398796\pi$
$3$		0			0
$3$		0			0
$4$	−5.21819			−2.60909
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	4.21819			1.59432			0.797162	−	0.603765i	$-0.206333\pi$
$7$	4.21819			1.59432			0.797162	+	0.603765i	$0.206333\pi$
$8$			8.64620i			3.05689i
$9$		0			0
$10$			2.68667i			0.849599i
$11$			2.31446i			0.697836i	0.937153	+	0.348918i	$0.113451\pi$
$11$			2.31446i			0.697836i	−0.937153	+	0.348918i	$0.886549\pi$
$12$		0			0
$13$	−0.643274			−0.178412			−0.0892061	−	0.996013i	$-0.528433\pi$
$13$	−0.643274			−0.178412			−0.0892061	+	0.996013i	$0.528433\pi$
$14$		−	11.3329i		−	3.02884i
$15$		0			0
$16$	12.7931			3.19827
$17$		−	0.586195i		−	0.142173i	−0.997470	−	0.0710866i	$-0.977353\pi$
$17$		−	0.586195i		−	0.142173i	0.997470	−	0.0710866i	$-0.0226467\pi$
$18$		0			0
$19$		−	5.95953i		−	1.36721i	−0.729852	−	0.683605i	$-0.760411\pi$
$19$		−	5.95953i		−	1.36721i	0.729852	−	0.683605i	$-0.239589\pi$
$20$	5.21819			1.16682
$21$		0			0
$22$	6.21819			1.32572
$23$	5.57491			1.16245			0.581225	−	0.813743i	$-0.302574\pi$
$23$	5.57491			1.16245			0.581225	+	0.813743i	$0.302574\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$			1.72826i			0.338941i
$27$		0			0
$28$	−22.0113			−4.15974
$29$	5.21819	−	1.33061i	0.968993	−	0.247088i
$29$	5.21819	−	1.33061i	0.968993	−	0.247088i
$30$		0			0
$31$		−	0.158221i		−	0.0284174i	−0.999899	−	0.0142087i	$-0.995477\pi$
$31$		−	0.158221i		−	0.0284174i	0.999899	−	0.0142087i	$-0.00452291\pi$
$32$		−	17.0784i		−	3.01907i
$33$		0			0
$34$	−1.57491			−0.270095
$35$	−4.21819			−0.713004
$36$		0			0
$37$		−	4.04272i		−	0.664620i	−0.943170	−	0.332310i	$-0.892172\pi$
$37$		−	4.04272i		−	0.664620i	0.943170	−	0.332310i	$-0.107828\pi$
$38$	−16.0113			−2.59737
$39$		0			0
$40$		−	8.64620i		−	1.36708i
$41$		−	9.76282i		−	1.52470i	−0.647167	−	0.762349i	$-0.724046\pi$
$41$		−	9.76282i		−	1.52470i	0.647167	−	0.762349i	$-0.275954\pi$
$42$		0			0
$43$		−	4.97568i		−	0.758785i	−0.925236	−	0.379392i	$-0.876133\pi$
$43$		−	4.97568i		−	0.758785i	0.925236	−	0.379392i	$-0.123867\pi$
$44$		−	12.0773i		−	1.82072i
$45$		0			0
$46$		−	14.9779i		−	2.20838i
$47$			2.31446i			0.337599i	0.985650	+	0.168799i	$0.0539890\pi$
$47$			2.31446i			0.337599i	−0.985650	+	0.168799i	$0.946011\pi$
$48$		0			0
$49$	10.7931			1.54187
$50$		−	2.68667i		−	0.379952i
$51$		0			0
$52$	3.35673			0.465494
$53$	−13.0796			−1.79663			−0.898314	−	0.439354i	$-0.855207\pi$
$53$	−13.0796			−1.79663			−0.898314	+	0.439354i	$0.855207\pi$
$54$		0			0
$55$		−	2.31446i		−	0.312082i
$56$			36.4713i			4.87368i
$57$		0			0
$58$	−3.57491	−	14.0195i	−0.469409	−	1.84086i
$59$	2.64327			0.344125			0.172063	−	0.985086i	$-0.444957\pi$
$59$	2.64327			0.344125			0.172063	+	0.985086i	$0.444957\pi$
$60$		0			0
$61$			0.983848i			0.125969i	0.998015	+	0.0629844i	$0.0200618\pi$
$61$			0.983848i			0.125969i	−0.998015	+	0.0629844i	$0.979938\pi$
$62$	−0.425088			−0.0539862
$63$		0			0
$64$	−20.2978			−2.53723
$65$	0.643274			0.0797884
$66$		0			0
$67$	1.57491			0.192406			0.0962031	−	0.995362i	$-0.469330\pi$
$67$	1.57491			0.192406			0.0962031	+	0.995362i	$0.469330\pi$
$68$			3.05888i			0.370943i
$69$		0			0
$70$			11.3329i			1.35454i
$71$	9.35673			1.11044			0.555220	−	0.831704i	$-0.312634\pi$
$71$	9.35673			1.11044			0.555220	+	0.831704i	$0.312634\pi$
$72$		0			0
$73$			9.17663i			1.07404i	0.843568	+	0.537022i	$0.180451\pi$
$73$			9.17663i			1.07404i	−0.843568	+	0.537022i	$0.819549\pi$
$74$	−10.8615			−1.26262
$75$		0			0
$76$			31.0980i			3.56718i
$77$			9.76282i			1.11258i
$78$		0			0
$79$		−	4.97568i		−	0.559808i	−0.960028	−	0.279904i	$-0.909697\pi$
$79$		−	4.97568i		−	0.559808i	0.960028	−	0.279904i	$-0.0903027\pi$
$80$	−12.7931			−1.43031
$81$		0			0
$82$	−26.2295			−2.89656
$83$	8.21819			0.902063			0.451032	−	0.892508i	$-0.351056\pi$
$83$	8.21819			0.902063			0.451032	+	0.892508i	$0.351056\pi$
$84$		0			0
$85$			0.586195i			0.0635818i
$86$	−13.3680			−1.44151
$87$		0			0
$88$	−20.0113			−2.13321
$89$			7.29014i			0.772754i	0.922341	+	0.386377i	$0.126274\pi$
$89$			7.29014i			0.772754i	−0.922341	+	0.386377i	$0.873726\pi$
$90$		0			0
$91$	−2.71345			−0.284447
$92$	−29.0909			−3.03294
$93$		0			0
$94$	6.21819			0.641357
$95$			5.95953i			0.611435i
$96$		0			0
$97$		−	3.05888i		−	0.310582i	−0.987869	−	0.155291i	$-0.950369\pi$
$97$		−	3.05888i		−	0.310582i	0.987869	−	0.155291i	$-0.0496315\pi$
$98$		−	28.9975i		−	2.92919i
$99$		0			0
$100$	−5.21819			−0.521819
$101$			1.48883i			0.148144i	0.997253	+	0.0740722i	$0.0235995\pi$
$101$			1.48883i			0.148144i	−0.997253	+	0.0740722i	$0.976400\pi$
$102$		0			0
$103$	11.2978			1.11321			0.556604	−	0.830778i	$-0.312104\pi$
$103$	11.2978			1.11321			0.556604	+	0.830778i	$0.312104\pi$
$104$		−	5.56188i		−	0.545387i
$105$		0			0
$106$			35.1407i			3.41316i
$107$	2.93164			0.283412			0.141706	−	0.989909i	$-0.454741\pi$
$107$	2.93164			0.283412			0.141706	+	0.989909i	$0.454741\pi$
$108$		0			0
$109$	−17.0796			−1.63593			−0.817967	−	0.575265i	$-0.804899\pi$
$109$	−17.0796			−1.63593			−0.817967	+	0.575265i	$0.804899\pi$
$110$	−6.21819			−0.592881
$111$		0			0
$112$	53.9637			5.09909
$113$		−	13.4891i		−	1.26895i	−0.772944	−	0.634474i	$-0.781217\pi$
$113$		−	13.4891i		−	1.26895i	0.772944	−	0.634474i	$-0.218783\pi$
$114$		0			0
$115$	−5.57491			−0.519863
$116$	−27.2295	+	6.94338i	−2.52819	+	0.644677i
$117$		0			0
$118$		−	7.10160i		−	0.653755i
$119$		−	2.47268i		−	0.226670i
$120$		0			0
$121$	5.64327			0.513025
$122$	2.64327			0.239311
$123$		0			0
$124$			0.825627i			0.0741435i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$			11.0934i			0.984383i	0.870487	+	0.492192i	$0.163804\pi$
$127$			11.0934i			0.984383i	−0.870487	+	0.492192i	$0.836196\pi$
$128$			20.3767i			1.80106i
$129$		0			0
$130$		−	1.72826i		−	0.151579i
$131$			7.13192i			0.623119i	0.950227	+	0.311559i	$0.100851\pi$
$131$			7.13192i			0.623119i	−0.950227	+	0.311559i	$0.899149\pi$
$132$		0			0
$133$		−	25.1384i		−	2.17978i
$134$		−	4.23127i		−	0.365526i
$135$		0			0
$136$	5.06836			0.434608
$137$		−	14.7894i		−	1.26354i	−0.775154	−	0.631772i	$-0.782328\pi$
$137$		−	14.7894i		−	1.26354i	0.775154	−	0.631772i	$-0.217672\pi$
$138$		0			0
$139$	3.56363			0.302263			0.151131	−	0.988514i	$-0.451708\pi$
$139$	3.56363			0.302263			0.151131	+	0.988514i	$0.451708\pi$
$140$	22.0113			1.86029
$141$		0			0
$142$		−	25.1384i		−	2.10957i
$143$		−	1.48883i		−	0.124502i
$144$		0			0
$145$	−5.21819	+	1.33061i	−0.433347	+	0.110501i
$146$	24.6546			2.04043
$147$		0			0
$148$			21.0957i			1.73406i
$149$	−6.00000			−0.491539			−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000			−0.491539			−0.245770	+	0.969328i	$0.579041\pi$
$150$		0			0
$151$	4.50655			0.366738			0.183369	−	0.983044i	$-0.441300\pi$
$151$	4.50655			0.366738			0.183369	+	0.983044i	$0.441300\pi$
$152$	51.5273			4.17942
$153$		0			0
$154$	26.2295			2.11363
$155$			0.158221i			0.0127086i
$156$		0			0
$157$		−	16.9456i		−	1.35241i	−0.736714	−	0.676205i	$-0.763624\pi$
$157$		−	16.9456i		−	1.35241i	0.736714	−	0.676205i	$-0.236376\pi$
$158$	−13.3680			−1.06350
$159$		0			0
$160$			17.0784i			1.35017i
$161$	23.5160			1.85332
$162$		0			0
$163$		−	11.0934i		−	0.868905i	−0.900695	−	0.434453i	$-0.856942\pi$
$163$		−	11.0934i		−	0.868905i	0.900695	−	0.434453i	$-0.143058\pi$
$164$			50.9442i			3.97808i
$165$		0			0
$166$		−	22.0795i		−	1.71370i
$167$	5.57491			0.431400			0.215700	−	0.976460i	$-0.430797\pi$
$167$	5.57491			0.431400			0.215700	+	0.976460i	$0.430797\pi$
$168$		0			0
$169$	−12.5862			−0.968169
$170$	1.57491			0.120790
$171$		0			0
$172$			25.9640i			1.97974i
$173$	−7.79310			−0.592498			−0.296249	−	0.955111i	$-0.595736\pi$
$173$	−7.79310			−0.592498			−0.296249	+	0.955111i	$0.595736\pi$
$174$		0			0
$175$	4.21819			0.318865
$176$			29.6091i			2.23187i
$177$		0			0
$178$	19.5862			1.46805
$179$	4.43637			0.331590			0.165795	−	0.986160i	$-0.446981\pi$
$179$	4.43637			0.331590			0.165795	+	0.986160i	$0.446981\pi$
$180$		0			0
$181$	13.5160			1.00464			0.502319	−	0.864682i	$-0.332480\pi$
$181$	13.5160			1.00464			0.502319	+	0.864682i	$0.332480\pi$
$182$			7.29014i			0.540381i
$183$		0			0
$184$			48.2018i			3.55348i
$185$			4.04272i			0.297227i
$186$		0			0
$187$	1.35673			0.0992136
$188$		−	12.0773i		−	0.880827i
$189$		0			0
$190$	16.0113			1.16158
$191$		−	11.5723i		−	0.837342i	−0.908138	−	0.418671i	$-0.862496\pi$
$191$		−	11.5723i		−	0.837342i	0.908138	−	0.418671i	$-0.137504\pi$
$192$		0			0
$193$			9.84404i			0.708589i	0.935134	+	0.354295i	$0.115279\pi$
$193$			9.84404i			0.708589i	−0.935134	+	0.354295i	$0.884721\pi$
$194$	−8.21819			−0.590031
$195$		0			0
$196$	−56.3204			−4.02289
$197$	5.14982			0.366910			0.183455	−	0.983028i	$-0.441272\pi$
$197$	5.14982			0.366910			0.183455	+	0.983028i	$0.441272\pi$
$198$		0			0
$199$	−17.7931			−1.26132			−0.630660	−	0.776060i	$-0.717216\pi$
$199$	−17.7931			−1.26132			−0.630660	+	0.776060i	$0.717216\pi$
$200$			8.64620i			0.611379i
$201$		0			0
$202$	4.00000			0.281439
$203$	22.0113	−	5.61277i	1.54489	−	0.393939i
$204$		0			0
$205$			9.76282i			0.681865i
$206$		−	30.3535i		−	2.11483i
$207$		0			0
$208$	−8.22947			−0.570611
$209$	13.7931			0.954089
$210$		0			0
$211$		−	18.8624i		−	1.29854i	−0.760556	−	0.649272i	$-0.775074\pi$
$211$		−	18.8624i		−	1.29854i	0.760556	−	0.649272i	$-0.224926\pi$
$212$	68.2520			4.68757
$213$		0			0
$214$		−	7.87634i		−	0.538415i
$215$			4.97568i			0.339339i
$216$		0			0
$217$		−	0.667406i		−	0.0453065i
$218$			45.8873i			3.10788i
$219$		0			0
$220$			12.0773i			0.814250i
$221$			0.377084i			0.0253654i
$222$		0			0
$223$	13.5749			0.909043			0.454522	−	0.890736i	$-0.349810\pi$
$223$	13.5749			0.909043			0.454522	+	0.890736i	$0.349810\pi$
$224$		−	72.0399i		−	4.81337i
$225$		0			0
$226$	−36.2408			−2.41070
$227$	−7.00181			−0.464727			−0.232363	−	0.972629i	$-0.574646\pi$
$227$	−7.00181			−0.464727			−0.232363	+	0.972629i	$0.574646\pi$
$228$		0			0
$229$			24.1546i			1.59618i	0.602539	+	0.798089i	$0.294156\pi$
$229$			24.1546i			1.59618i	−0.602539	+	0.798089i	$0.705844\pi$
$230$			14.9779i			0.987616i
$231$		0			0
$232$	11.5047	+	45.1175i	0.755323	+	2.96211i
$233$	−5.14982			−0.337376			−0.168688	−	0.985669i	$-0.553953\pi$
$233$	−5.14982			−0.337376			−0.168688	+	0.985669i	$0.553953\pi$
$234$		0			0
$235$		−	2.31446i		−	0.150979i
$236$	−13.7931			−0.897854
$237$		0			0
$238$	−6.64327			−0.430620
$239$	−14.6433			−0.947195			−0.473597	−	0.880741i	$-0.657045\pi$
$239$	−14.6433			−0.947195			−0.473597	+	0.880741i	$0.657045\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		−	15.1616i		−	0.974625i
$243$		0			0
$244$		−	5.13390i		−	0.328665i
$245$	−10.7931			−0.689546
$246$		0			0
$247$			3.83361i			0.243927i
$248$	1.36801			0.0868688
$249$		0			0
$250$			2.68667i			0.169920i
$251$			22.6354i			1.42873i	0.699771	+	0.714367i	$0.253285\pi$
$251$			22.6354i			1.42873i	−0.699771	+	0.714367i	$0.746715\pi$
$252$		0			0
$253$			12.9029i			0.811199i
$254$	29.8044			1.87009
$255$		0			0
$256$	14.1498			0.884364
$257$	−10.9204			−0.681193			−0.340596	−	0.940210i	$-0.610629\pi$
$257$	−10.9204			−0.681193			−0.340596	+	0.940210i	$0.610629\pi$
$258$		0			0
$259$		−	17.0530i		−	1.05962i
$260$	−3.35673			−0.208175
$261$		0			0
$262$	19.1611			1.18378
$263$		−	9.92105i		−	0.611758i	−0.952070	−	0.305879i	$-0.901050\pi$
$263$		−	9.92105i		−	0.611758i	0.952070	−	0.305879i	$-0.0989503\pi$
$264$		0			0
$265$	13.0796			0.803476
$266$	−67.5386			−4.14106
$267$		0			0
$268$	−8.21819			−0.502006
$269$			12.4240i			0.757508i	0.925497	+	0.378754i	$0.123647\pi$
$269$			12.4240i			0.757508i	−0.925497	+	0.378754i	$0.876353\pi$
$270$		0			0
$271$		−	27.2643i		−	1.65619i	−0.560587	−	0.828095i	$-0.689425\pi$
$271$		−	27.2643i		−	1.65619i	0.560587	−	0.828095i	$-0.310575\pi$
$272$		−	7.49925i		−	0.454709i
$273$		0			0
$274$	−39.7342			−2.40043
$275$			2.31446i			0.139567i
$276$		0			0
$277$	−2.92035			−0.175467			−0.0877335	−	0.996144i	$-0.527962\pi$
$277$	−2.92035			−0.175467			−0.0877335	+	0.996144i	$0.527962\pi$
$278$		−	9.57428i		−	0.574227i
$279$		0			0
$280$		−	36.4713i		−	2.17958i
$281$	18.3662			1.09564			0.547818	−	0.836598i	$-0.315459\pi$
$281$	18.3662			1.09564			0.547818	+	0.836598i	$0.315459\pi$
$282$		0			0
$283$	−14.8615			−0.883422			−0.441711	−	0.897157i	$-0.645628\pi$
$283$	−14.8615			−0.883422			−0.441711	+	0.897157i	$0.645628\pi$
$284$	−48.8251			−2.89724
$285$		0			0
$286$	−4.00000			−0.236525
$287$		−	41.1814i		−	2.43086i
$288$		0			0
$289$	16.6564			0.979787
$290$	3.57491	+	14.0195i	0.209926	+	0.823256i
$291$		0			0
$292$		−	47.8854i		−	2.80228i
$293$		−	1.57004i		−	0.0917229i	−0.998948	−	0.0458615i	$-0.985397\pi$
$293$		−	1.57004i		−	0.0917229i	0.998948	−	0.0458615i	$-0.0146033\pi$
$294$		0			0
$295$	−2.64327			−0.153897
$296$	34.9542			2.03167
$297$		0			0
$298$			16.1200i			0.933807i
$299$	−3.58620			−0.207395
$300$		0			0
$301$		−	20.9884i		−	1.20975i
$302$		−	12.1076i		−	0.696714i
$303$		0			0
$304$		−	76.2409i		−	4.37271i
$305$		−	0.983848i		−	0.0563350i
$306$		0			0
$307$			4.65924i			0.265917i	0.991122	+	0.132958i	$0.0424477\pi$
$307$			4.65924i			0.265917i	−0.991122	+	0.132958i	$0.957552\pi$
$308$		−	50.9442i		−	2.90282i
$309$		0			0
$310$	0.425088			0.0241434
$311$			21.6516i			1.22775i	0.789404	+	0.613874i	$0.210390\pi$
$311$			21.6516i			1.22775i	−0.789404	+	0.613874i	$0.789610\pi$
$312$		0			0
$313$	11.3567			0.641920			0.320960	−	0.947093i	$-0.395994\pi$
$313$	11.3567			0.641920			0.320960	+	0.947093i	$0.395994\pi$
$314$	−45.5273			−2.56925
$315$		0			0
$316$			25.9640i			1.46059i
$317$			9.36517i			0.526000i	0.964796	+	0.263000i	$0.0847120\pi$
$317$			9.36517i			0.526000i	−0.964796	+	0.263000i	$0.915288\pi$
$318$		0			0
$319$	3.07965	+	12.0773i	0.172427	+	0.676198i
$320$	20.2978			1.13468
$321$		0			0
$322$		−	63.1797i		−	3.52087i
$323$	−3.49345			−0.194381
$324$		0			0
$325$	−0.643274			−0.0356824
$326$	−29.8044			−1.65071
$327$		0			0
$328$	84.4113			4.66084
$329$			9.76282i			0.538242i
$330$		0			0
$331$			18.5115i			1.01748i	0.860919	+	0.508741i	$0.169889\pi$
$331$			18.5115i			1.01748i	−0.860919	+	0.508741i	$0.830111\pi$
$332$	−42.8840			−2.35357
$333$		0			0
$334$		−	14.9779i		−	0.819556i
$335$	−1.57491			−0.0860467
$336$		0			0
$337$			18.8116i			1.02473i	0.858768	+	0.512365i	$0.171231\pi$
$337$			18.8116i			1.02473i	−0.858768	+	0.512365i	$0.828769\pi$
$338$			33.8149i			1.83929i
$339$		0			0
$340$		−	3.05888i		−	0.165891i
$341$	0.366196			0.0198306
$342$		0			0
$343$	16.0000			0.863919
$344$	43.0208			2.31952
$345$		0			0
$346$			20.9375i			1.12561i
$347$	25.2313			1.35449			0.677243	−	0.735759i	$-0.263174\pi$
$347$	25.2313			1.35449			0.677243	+	0.735759i	$0.263174\pi$
$348$		0			0
$349$	2.85018			0.152566			0.0762832	−	0.997086i	$-0.475695\pi$
$349$	2.85018			0.152566			0.0762832	+	0.997086i	$0.475695\pi$
$350$		−	11.3329i		−	0.605767i
$351$		0			0
$352$	39.5273			2.10681
$353$	30.0927			1.60168			0.800838	−	0.598881i	$-0.204388\pi$
$353$	30.0927			1.60168			0.800838	+	0.598881i	$0.204388\pi$
$354$		0			0
$355$	−9.35673			−0.496603
$356$		−	38.0413i		−	2.01619i
$357$		0			0
$358$		−	11.9191i		−	0.629942i
$359$			23.6193i			1.24658i	0.781992	+	0.623289i	$0.214204\pi$
$359$			23.6193i			1.24658i	−0.781992	+	0.623289i	$0.785796\pi$
$360$		0			0
$361$	−16.5160			−0.869264
$362$		−	36.3131i		−	1.90857i
$363$		0			0
$364$	14.1593			0.742149
$365$		−	9.17663i		−	0.480327i
$366$		0			0
$367$			17.2112i			0.898417i	0.893427	+	0.449208i	$0.148294\pi$
$367$			17.2112i			0.898417i	−0.893427	+	0.449208i	$0.851706\pi$
$368$	71.3204			3.71783
$369$		0			0
$370$	10.8615			0.564660
$371$	−55.1724			−2.86441
$372$		0			0
$373$	22.8727			1.18431			0.592153	−	0.805826i	$-0.298278\pi$
$373$	22.8727			1.18431			0.592153	+	0.805826i	$0.298278\pi$
$374$		−	3.64507i		−	0.188482i
$375$		0			0
$376$	−20.0113			−1.03200
$377$	−3.35673	+	0.855948i	−0.172880	+	0.0440836i
$378$		0			0
$379$			27.2643i			1.40047i	0.713910	+	0.700237i	$0.246923\pi$
$379$			27.2643i			1.40047i	−0.713910	+	0.700237i	$0.753077\pi$
$380$		−	31.0980i		−	1.59529i
$381$		0			0
$382$	−31.0909			−1.59075
$383$	−37.2313			−1.90243			−0.951215	−	0.308529i	$-0.900163\pi$
$383$	−37.2313			−1.90243			−0.951215	+	0.308529i	$0.900163\pi$
$384$		0			0
$385$		−	9.76282i		−	0.497560i
$386$	26.4477			1.34615
$387$		0			0
$388$			15.9618i			0.810337i
$389$		−	23.6496i		−	1.19908i	−0.800344	−	0.599541i	$-0.795350\pi$
$389$		−	23.6496i		−	1.19908i	0.800344	−	0.599541i	$-0.204650\pi$
$390$		0			0
$391$		−	3.26799i		−	0.165269i
$392$			93.3193i			4.71334i
$393$		0			0
$394$		−	13.8359i		−	0.697041i
$395$			4.97568i			0.250354i
$396$		0			0
$397$	−30.0226			−1.50679			−0.753395	−	0.657568i	$-0.771585\pi$
$397$	−30.0226			−1.50679			−0.753395	+	0.657568i	$0.771585\pi$
$398$			47.8042i			2.39621i
$399$		0			0
$400$	12.7931			0.639655
$401$	32.3698			1.61647			0.808236	−	0.588859i	$-0.200423\pi$
$401$	32.3698			1.61647			0.808236	+	0.588859i	$0.200423\pi$
$402$		0			0
$403$			0.101780i			0.00507000i
$404$		−	7.76901i		−	0.386523i
$405$		0			0
$406$	−15.0796	−	59.1370i	−0.748390	−	2.93492i
$407$	9.35673			0.463796
$408$		0			0
$409$			24.5055i			1.21172i	0.795571	+	0.605860i	$0.207171\pi$
$409$			24.5055i			1.21172i	−0.795571	+	0.605860i	$0.792829\pi$
$410$	26.2295			1.29538
$411$		0			0
$412$	−58.9542			−2.90447
$413$	11.1498			0.548647
$414$		0			0
$415$	−8.21819			−0.403415
$416$			10.9861i			0.538638i
$417$		0			0
$418$		−	37.0575i		−	1.81254i
$419$	−6.22947			−0.304330			−0.152165	−	0.988355i	$-0.548624\pi$
$419$	−6.22947			−0.304330			−0.152165	+	0.988355i	$0.548624\pi$
$420$		0			0
$421$			32.9074i			1.60381i	0.597452	+	0.801905i	$0.296180\pi$
$421$			32.9074i			1.60381i	−0.597452	+	0.801905i	$0.703820\pi$
$422$	−50.6771			−2.46692
$423$		0			0
$424$		−	113.089i		−	5.49210i
$425$		−	0.586195i		−	0.0284346i
$426$		0			0
$427$			4.15006i			0.200835i
$428$	−15.2978			−0.739449
$429$		0			0
$430$	13.3680			0.644663
$431$	−3.00947			−0.144961			−0.0724806	−	0.997370i	$-0.523092\pi$
$431$	−3.00947			−0.144961			−0.0724806	+	0.997370i	$0.523092\pi$
$432$		0			0
$433$			34.3150i			1.64908i	0.565807	+	0.824538i	$0.308565\pi$
$433$			34.3150i			1.64908i	−0.565807	+	0.824538i	$0.691435\pi$
$434$	−1.79310			−0.0860715
$435$		0			0
$436$	89.1248			4.26830
$437$		−	33.2239i		−	1.58931i
$438$		0			0
$439$	29.8157			1.42302			0.711512	−	0.702674i	$-0.248011\pi$
$439$	29.8157			1.42302			0.711512	+	0.702674i	$0.248011\pi$
$440$	20.0113			0.954001
$441$		0			0
$442$	1.01310			0.0481883
$443$		−	27.9579i		−	1.32832i	−0.747591	−	0.664159i	$-0.768790\pi$
$443$		−	27.9579i		−	1.32832i	0.747591	−	0.664159i	$-0.231210\pi$
$444$		0			0
$445$		−	7.29014i		−	0.345586i
$446$		−	36.4713i		−	1.72697i
$447$		0			0
$448$	−85.6201			−4.04517
$449$		−	4.27796i		−	0.201889i	−0.994892	−	0.100945i	$-0.967814\pi$
$449$		−	4.27796i		−	0.201889i	0.994892	−	0.100945i	$-0.0321865\pi$
$450$		0			0
$451$	22.5957			1.06399
$452$			70.3887i			3.31081i
$453$		0			0
$454$			18.8116i			0.882870i
$455$	2.71345			0.127209
$456$		0			0
$457$	−10.3662			−0.484910			−0.242455	−	0.970163i	$-0.577953\pi$
$457$	−10.3662			−0.484910			−0.242455	+	0.970163i	$0.577953\pi$
$458$	64.8953			3.03236
$459$		0			0
$460$	29.0909			1.35637
$461$			13.0915i			0.609730i	0.952396	+	0.304865i	$0.0986113\pi$
$461$			13.0915i			0.609730i	−0.952396	+	0.304865i	$0.901389\pi$
$462$		0			0
$463$	−33.4571			−1.55488			−0.777442	−	0.628954i	$-0.783483\pi$
$463$	−33.4571			−1.55488			−0.777442	+	0.628954i	$0.783483\pi$
$464$	66.7568	−	17.0226i	3.09911	−	0.790257i
$465$		0			0
$466$			13.8359i			0.640934i
$467$			32.5868i			1.50794i	0.656911	+	0.753968i	$0.271863\pi$
$467$			32.5868i			1.50794i	−0.656911	+	0.753968i	$0.728137\pi$
$468$		0			0
$469$	6.64327			0.306758
$470$	−6.21819			−0.286824
$471$		0			0
$472$			22.8543i			1.05195i
$473$	11.5160			0.529507
$474$		0			0
$475$		−	5.95953i		−	0.273442i
$476$			12.9029i			0.591404i
$477$		0			0
$478$			39.3416i			1.79944i
$479$			36.5222i			1.66874i	0.551204	+	0.834370i	$0.314169\pi$
$479$			36.5222i			1.66874i	−0.551204	+	0.834370i	$0.685831\pi$
$480$		0			0
$481$			2.60058i			0.118576i
$482$		−	5.37334i		−	0.244749i
$483$		0			0
$484$	−29.4477			−1.33853
$485$			3.05888i			0.138896i
$486$		0			0
$487$	25.9411			1.17550			0.587752	−	0.809041i	$-0.300013\pi$
$487$	25.9411			1.17550			0.587752	+	0.809041i	$0.300013\pi$
$488$	−8.50655			−0.385073
$489$		0			0
$490$			28.9975i			1.30997i
$491$		−	26.3411i		−	1.18876i	−0.804185	−	0.594379i	$-0.797398\pi$
$491$		−	26.3411i		−	1.18876i	0.804185	−	0.594379i	$-0.202602\pi$
$492$		0			0
$493$	−0.779998	−	3.05888i	−0.0351294	−	0.137765i
$494$	10.2996			0.463403
$495$		0			0
$496$		−	2.02414i		−	0.0908865i
$497$	39.4684			1.77040
$498$		0			0
$499$	−21.0131			−0.940676			−0.470338	−	0.882486i	$-0.655868\pi$
$499$	−21.0131			−0.940676			−0.470338	+	0.882486i	$0.655868\pi$
$500$	5.21819			0.233364
$501$		0			0
$502$	60.8139			2.71426
$503$			14.5500i			0.648751i	0.945928	+	0.324375i	$0.105154\pi$
$503$			14.5500i			0.648751i	−0.945928	+	0.324375i	$0.894846\pi$
$504$		0			0
$505$		−	1.48883i		−	0.0662522i
$506$	34.6658			1.54108
$507$		0			0
$508$		−	57.8876i		−	2.56835i
$509$	12.2295			0.542062			0.271031	−	0.962571i	$-0.412635\pi$
$509$	12.2295			0.542062			0.271031	+	0.962571i	$0.412635\pi$
$510$		0			0
$511$			38.7087i			1.71237i
$512$			2.73756i			0.120984i
$513$		0			0
$514$			29.3394i			1.29410i
$515$	−11.2978			−0.497842
$516$		0			0
$517$	−5.35673			−0.235589
$518$	−45.8157			−2.01302
$519$		0			0
$520$			5.56188i			0.243905i
$521$	−17.5160			−0.767391			−0.383695	−	0.923460i	$-0.625349\pi$
$521$	−17.5160			−0.767391			−0.383695	+	0.923460i	$0.625349\pi$
$522$		0			0
$523$	−10.8840			−0.475926			−0.237963	−	0.971274i	$-0.576480\pi$
$523$	−10.8840			−0.475926			−0.237963	+	0.971274i	$0.576480\pi$
$524$		−	37.2157i		−	1.62578i
$525$		0			0
$526$	−26.6546			−1.16219
$527$	−0.0927485			−0.00404019
$528$		0			0
$529$	8.07965			0.351289
$530$		−	35.1407i		−	1.52641i
$531$		0			0
$532$			131.177i			5.68724i
$533$			6.28017i			0.272025i
$534$		0			0
$535$	−2.93164			−0.126746
$536$			13.6170i			0.588165i
$537$		0			0
$538$	33.3793			1.43908
$539$			24.9802i			1.07597i
$540$		0			0
$541$		−	35.7572i		−	1.53732i	−0.639657	−	0.768661i	$-0.720923\pi$
$541$		−	35.7572i		−	1.53732i	0.639657	−	0.768661i	$-0.279077\pi$
$542$	−73.2502			−3.14637
$543$		0			0
$544$	−10.0113			−0.429230
$545$	17.0796			0.731612
$546$		0			0
$547$	12.6320			0.540105			0.270052	−	0.962846i	$-0.412959\pi$
$547$	12.6320			0.540105			0.270052	+	0.962846i	$0.412959\pi$
$548$			77.1738i			3.29670i
$549$		0			0
$550$	6.21819			0.265144
$551$	−7.92982	−	31.0980i	−0.337822	−	1.32482i
$552$		0			0
$553$		−	20.9884i		−	0.892516i
$554$			7.84602i			0.333345i
$555$		0			0
$556$	−18.5957			−0.788632
$557$	21.0095			0.890200			0.445100	−	0.895481i	$-0.353168\pi$
$557$	21.0095			0.890200			0.445100	+	0.895481i	$0.353168\pi$
$558$		0			0
$559$			3.20073i			0.135376i
$560$	−53.9637			−2.28038
$561$		0			0
$562$		−	49.3439i		−	2.08145i
$563$		−	45.6782i		−	1.92511i	−0.271090	−	0.962554i	$-0.587384\pi$
$563$		−	45.6782i		−	1.92511i	0.271090	−	0.962554i	$-0.412616\pi$
$564$		0			0
$565$			13.4891i			0.567491i
$566$			39.9278i			1.67829i
$567$		0			0
$568$			80.9001i			3.39449i
$569$			38.5463i			1.61595i	0.589220	+	0.807973i	$0.299435\pi$
$569$			38.5463i			1.61595i	−0.589220	+	0.807973i	$0.700565\pi$
$570$		0			0
$571$	−25.7229			−1.07647			−0.538235	−	0.842795i	$-0.680909\pi$
$571$	−25.7229			−1.07647			−0.538235	+	0.842795i	$0.680909\pi$
$572$			7.76901i			0.324839i
$573$		0			0
$574$	−110.641			−4.61806
$575$	5.57491			0.232490
$576$		0			0
$577$		−	31.7145i		−	1.32029i	−0.751138	−	0.660145i	$-0.770495\pi$
$577$		−	31.7145i		−	1.32029i	0.751138	−	0.660145i	$-0.229505\pi$
$578$		−	44.7502i		−	1.86136i
$579$		0			0
$580$	27.2295	−	6.94338i	1.13064	−	0.288308i
$581$	34.6658			1.43818
$582$		0			0
$583$		−	30.2723i		−	1.25375i
$584$	−79.3430			−3.28324
$585$		0			0
$586$	−4.21819			−0.174252
$587$	−37.5975			−1.55181			−0.775907	−	0.630847i	$-0.782707\pi$
$587$	−37.5975			−1.55181			−0.775907	+	0.630847i	$0.782707\pi$
$588$		0			0
$589$	−0.942924			−0.0388525
$590$			7.10160i			0.292368i
$591$		0			0
$592$		−	51.7190i		−	2.12564i
$593$	−36.2295			−1.48777			−0.743883	−	0.668310i	$-0.767018\pi$
$593$	−36.2295			−1.48777			−0.743883	+	0.668310i	$0.767018\pi$
$594$		0			0
$595$			2.47268i			0.101370i
$596$	31.3091			1.28247
$597$		0			0
$598$			9.63492i			0.394001i
$599$		−	3.48685i		−	0.142469i	−0.997460	−	0.0712344i	$-0.977306\pi$
$599$		−	3.48685i		−	0.142469i	0.997460	−	0.0712344i	$-0.0226938\pi$
$600$		0			0
$601$			4.81746i			0.196508i	0.995161	+	0.0982542i	$0.0313258\pi$
$601$			4.81746i			0.196508i	−0.995161	+	0.0982542i	$0.968674\pi$
$602$	−56.3888			−2.29823
$603$		0			0
$604$	−23.5160			−0.956853
$605$	−5.64327			−0.229432
$606$		0			0
$607$			7.57626i			0.307511i	0.988109	+	0.153756i	$0.0491368\pi$
$607$			7.57626i			0.307511i	−0.988109	+	0.153756i	$0.950863\pi$
$608$	−101.779			−4.12770
$609$		0			0
$610$	−2.64327			−0.107023
$611$		−	1.48883i		−	0.0602317i
$612$		0			0
$613$	4.15930			0.167992			0.0839962	−	0.996466i	$-0.473232\pi$
$613$	4.15930			0.167992			0.0839962	+	0.996466i	$0.473232\pi$
$614$	12.5178			0.505179
$615$		0			0
$616$	−84.4113			−3.40103
$617$		−	25.7246i		−	1.03563i	−0.855491	−	0.517817i	$-0.826745\pi$
$617$		−	25.7246i		−	1.03563i	0.855491	−	0.517817i	$-0.173255\pi$
$618$		0			0
$619$			31.1997i			1.25402i	0.779010	+	0.627012i	$0.215722\pi$
$619$			31.1997i			1.25402i	−0.779010	+	0.627012i	$0.784278\pi$
$620$		−	0.825627i		−	0.0331580i
$621$		0			0
$622$	58.1706			2.33243
$623$			30.7512i			1.23202i
$624$		0			0
$625$	1.00000			0.0400000
$626$		−	30.5118i		−	1.21949i
$627$		0			0
$628$			88.4255i			3.52856i
$629$	−2.36983			−0.0944912
$630$		0			0
$631$	22.1593			0.882148			0.441074	−	0.897471i	$-0.354598\pi$
$631$	22.1593			0.882148			0.441074	+	0.897471i	$0.354598\pi$
$632$	43.0208			1.71127
$633$		0			0
$634$	25.1611			0.999275
$635$		−	11.0934i		−	0.440230i
$636$		0			0
$637$	−6.94292			−0.275089
$638$	32.4477	−	8.27399i	1.28462	−	0.327570i
$639$		0			0
$640$		−	20.3767i		−	0.805461i
$641$			22.3754i			0.883776i	0.897070	+	0.441888i	$0.145691\pi$
$641$			22.3754i			0.883776i	−0.897070	+	0.441888i	$0.854309\pi$
$642$		0			0
$643$	−20.1480			−0.794560			−0.397280	−	0.917697i	$-0.630046\pi$
$643$	−20.1480			−0.794560			−0.397280	+	0.917697i	$0.630046\pi$
$644$	−122.711			−4.83549
$645$		0			0
$646$			9.38574i			0.369277i
$647$	−34.9542			−1.37419			−0.687096	−	0.726567i	$-0.741115\pi$
$647$	−34.9542			−1.37419			−0.687096	+	0.726567i	$0.741115\pi$
$648$		0			0
$649$			6.11775i			0.240143i
$650$			1.72826i			0.0677881i
$651$		0			0
$652$			57.8876i			2.26705i
$653$		−	17.3227i		−	0.677890i	−0.940806	−	0.338945i	$-0.889930\pi$
$653$		−	17.3227i		−	0.677890i	0.940806	−	0.338945i	$-0.110070\pi$
$654$		0			0
$655$		−	7.13192i		−	0.278667i
$656$		−	124.897i		−	4.87640i
$657$		0			0
$658$	26.2295			1.02253
$659$			17.1851i			0.669435i	0.942318	+	0.334718i	$0.108641\pi$
$659$			17.1851i			0.669435i	−0.942318	+	0.334718i	$0.891359\pi$
$660$		0			0
$661$	24.0891			0.936958			0.468479	−	0.883475i	$-0.344802\pi$
$661$	24.0891			0.936958			0.468479	+	0.883475i	$0.344802\pi$
$662$	49.7342			1.93297
$663$		0			0
$664$			71.0561i			2.75751i
$665$			25.1384i			0.974826i
$666$		0			0
$667$	29.0909	−	7.41804i	1.12641	−	0.287228i
$668$	−29.0909			−1.12556
$669$		0			0
$670$			4.23127i			0.163468i
$671$	−2.27708			−0.0879056
$672$		0			0
$673$	−31.4495			−1.21229			−0.606144	−	0.795355i	$-0.707285\pi$
$673$	−31.4495			−1.21229			−0.606144	+	0.795355i	$0.707285\pi$
$674$	50.5404			1.94674
$675$		0			0
$676$	65.6771			2.52604
$677$			29.0187i			1.11528i	0.830083	+	0.557640i	$0.188293\pi$
$677$			29.0187i			1.11528i	−0.830083	+	0.557640i	$0.811707\pi$
$678$		0			0
$679$		−	12.9029i		−	0.495168i
$680$	−5.06836			−0.194363
$681$		0			0
$682$		−	0.983848i		−	0.0376735i
$683$	−12.2884			−0.470201			−0.235101	−	0.971971i	$-0.575542\pi$
$683$	−12.2884			−0.470201			−0.235101	+	0.971971i	$0.575542\pi$
$684$		0			0
$685$			14.7894i			0.565074i
$686$		−	42.9867i		−	1.64124i
$687$		0			0
$688$		−	63.6544i		−	2.42680i
$689$	8.41380			0.320540
$690$		0			0
$691$	−41.6753			−1.58540			−0.792702	−	0.609609i	$-0.791326\pi$
$691$	−41.6753			−1.58540			−0.792702	+	0.609609i	$0.791326\pi$
$692$	40.6658			1.54588
$693$		0			0
$694$		−	67.7881i		−	2.57320i
$695$	−3.56363			−0.135176
$696$		0			0
$697$	−5.72292			−0.216771
$698$		−	7.65748i		−	0.289840i
$699$		0			0
$700$	−22.0113			−0.831948
$701$	−27.8157			−1.05058			−0.525292	−	0.850922i	$-0.676044\pi$
$701$	−27.8157			−1.05058			−0.525292	+	0.850922i	$0.676044\pi$
$702$		0			0
$703$	−24.0927			−0.908675
$704$		−	46.9785i		−	1.77057i
$705$		0			0
$706$		−	80.8492i		−	3.04280i
$707$			6.28017i			0.236190i
$708$		0			0
$709$	−0.643274			−0.0241587			−0.0120793	−	0.999927i	$-0.503845\pi$
$709$	−0.643274			−0.0241587			−0.0120793	+	0.999927i	$0.503845\pi$
$710$			25.1384i			0.943428i
$711$		0			0
$712$	−63.0320			−2.36223
$713$		−	0.882069i		−	0.0330337i
$714$		0			0
$715$			1.48883i			0.0556792i
$716$	−23.1498			−0.865150
$717$		0			0
$718$	63.4571			2.36820
$719$	−21.2389			−0.792079			−0.396039	−	0.918233i	$-0.629616\pi$
$719$	−21.2389			−0.792079			−0.396039	+	0.918233i	$0.629616\pi$
$720$		0			0
$721$	47.6564			1.77482
$722$			44.3731i			1.65139i
$723$		0			0
$724$	−70.5291			−2.62119
$725$	5.21819	−	1.33061i	0.193799	−	0.0494177i
$726$		0			0
$727$		−	19.0771i		−	0.707531i	−0.935334	−	0.353765i	$-0.884901\pi$
$727$		−	19.0771i		−	0.707531i	0.935334	−	0.353765i	$-0.115099\pi$
$728$		−	23.4610i		−	0.869524i
$729$		0			0
$730$	−24.6546			−0.912506
$731$	−2.91672			−0.107879
$732$		0			0
$733$		−	39.4835i		−	1.45836i	−0.684324	−	0.729178i	$-0.739903\pi$
$733$		−	39.4835i		−	1.45836i	0.684324	−	0.729178i	$-0.260097\pi$
$734$	46.2408			1.70678
$735$		0			0
$736$		−	95.2107i		−	3.50951i
$737$			3.64507i			0.134268i
$738$		0			0
$739$		−	5.95953i		−	0.219225i	−0.993974	−	0.109612i	$-0.965039\pi$
$739$		−	5.95953i		−	0.219225i	0.993974	−	0.109612i	$-0.0349610\pi$
$740$		−	21.0957i		−	0.775493i
$741$		0			0
$742$			148.230i			5.44169i
$743$		−	18.0065i		−	0.660594i	−0.943877	−	0.330297i	$-0.892851\pi$
$743$		−	18.0065i		−	0.660594i	0.943877	−	0.330297i	$-0.107149\pi$
$744$		0			0
$745$	6.00000			0.219823
$746$		−	61.4515i		−	2.24990i
$747$		0			0
$748$	−7.07965			−0.258858
$749$	12.3662			0.451851
$750$		0			0
$751$			25.8623i			0.943727i	0.881671	+	0.471864i	$0.156419\pi$
$751$			25.8623i			0.943727i	−0.881671	+	0.471864i	$0.843581\pi$
$752$			29.6091i			1.07973i
$753$		0			0
$754$	2.29965	+	9.01841i	0.0837483	+	0.328431i
$755$	−4.50655			−0.164010
$756$		0			0
$757$			53.0193i			1.92702i	0.267676	+	0.963509i	$0.413744\pi$
$757$			53.0193i			1.92702i	−0.267676	+	0.963509i	$0.586256\pi$
$758$	73.2502			2.66057
$759$		0			0
$760$	−51.5273			−1.86909
$761$	34.8953			1.26495			0.632477	−	0.774579i	$-0.282038\pi$
$761$	34.8953			1.26495			0.632477	+	0.774579i	$0.282038\pi$
$762$		0			0
$763$	−72.0451			−2.60821
$764$			60.3864i			2.18470i
$765$		0			0
$766$			100.028i			3.61416i
$767$	−1.70035			−0.0613961
$768$		0			0
$769$			30.5888i			1.10306i	0.834155	+	0.551530i	$0.185956\pi$
$769$			30.5888i			1.10306i	−0.834155	+	0.551530i	$0.814044\pi$
$770$	−26.2295			−0.945245
$771$		0			0
$772$		−	51.3680i		−	1.84878i
$773$			11.9658i			0.430378i	0.976572	+	0.215189i	$0.0690368\pi$
$773$			11.9658i			0.430378i	−0.976572	+	0.215189i	$0.930963\pi$
$774$		0			0
$775$		−	0.158221i		−	0.00568347i
$776$	26.4477			0.949416
$777$		0			0
$778$	−63.5386			−2.27797
$779$	−58.1819			−2.08458
$780$		0			0
$781$			21.6558i			0.774904i
$782$	−8.78000			−0.313972
$783$		0			0
$784$	138.077			4.93133
$785$			16.9456i			0.604816i
$786$		0			0
$787$	−21.2087			−0.756009			−0.378005	−	0.925804i	$-0.623390\pi$
$787$	−21.2087			−0.756009			−0.378005	+	0.925804i	$0.623390\pi$
$788$	−26.8727			−0.957302
$789$		0			0
$790$	13.3680			0.475613
$791$		−	56.8996i		−	2.02312i
$792$		0			0
$793$		−	0.632884i		−	0.0224744i
$794$			80.6607i			2.86254i
$795$		0			0
$796$	92.8477			3.29090
$797$		−	8.32068i		−	0.294734i	−0.989082	−	0.147367i	$-0.952920\pi$
$797$		−	8.32068i		−	0.294734i	0.989082	−	0.147367i	$-0.0470798\pi$
$798$		0			0
$799$	1.35673			0.0479975
$800$		−	17.0784i		−	0.603813i
$801$		0			0
$802$		−	86.9670i		−	3.07091i
$803$	−21.2389			−0.749506
$804$		0			0
$805$	−23.5160			−0.828831
$806$	0.273448			0.00963179
$807$		0			0
$808$	−12.8727			−0.452862
$809$			23.4610i			0.824846i	0.910992	+	0.412423i	$0.135317\pi$
$809$			23.4610i			0.824846i	−0.910992	+	0.412423i	$0.864683\pi$
$810$		0			0
$811$	−7.00947			−0.246136			−0.123068	−	0.992398i	$-0.539273\pi$
$811$	−7.00947			−0.246136			−0.123068	+	0.992398i	$0.539273\pi$
$812$	−114.859	+	29.2885i	−4.03076	+	1.02782i
$813$		0			0
$814$		−	25.1384i		−	0.881101i
$815$			11.0934i			0.388586i
$816$		0			0
$817$	−29.6527			−1.03742
$818$	65.8382			2.30198
$819$		0			0
$820$		−	50.9442i		−	1.77905i
$821$	−1.44584			−0.0504603			−0.0252302	−	0.999682i	$-0.508032\pi$
$821$	−1.44584			−0.0504603			−0.0252302	+	0.999682i	$0.508032\pi$
$822$		0			0
$823$			47.1671i			1.64414i	0.569386	+	0.822070i	$0.307181\pi$
$823$			47.1671i			1.64414i	−0.569386	+	0.822070i	$0.692819\pi$
$824$			97.6833i			3.40296i
$825$		0			0
$826$		−	29.9559i		−	1.04230i
$827$			20.1366i			0.700219i	0.936709	+	0.350109i	$0.113856\pi$
$827$			20.1366i			0.700219i	−0.936709	+	0.350109i	$0.886144\pi$
$828$		0			0
$829$			12.8011i			0.444602i	0.974978	+	0.222301i	$0.0713567\pi$
$829$			12.8011i			0.444602i	−0.974978	+	0.222301i	$0.928643\pi$
$830$			22.0795i			0.766392i
$831$		0			0
$832$	13.0571			0.452673
$833$		−	6.32686i		−	0.219213i
$834$		0			0
$835$	−5.57491			−0.192928
$836$	−71.9750			−2.48931
$837$		0			0
$838$			16.7365i			0.578154i
$839$			16.8341i			0.581178i	0.956848	+	0.290589i	$0.0938512\pi$
$839$			16.8341i			0.581178i	−0.956848	+	0.290589i	$0.906149\pi$
$840$		0			0
$841$	25.4589	−	13.8868i	0.877895	−	0.478854i
$842$	88.4113			3.04686
$843$		0			0
$844$			98.4278i			3.38802i
$845$	12.5862			0.432978
$846$		0			0
$847$	23.8044			0.817928
$848$	−167.329			−5.74611
$849$		0			0
$850$	−1.57491			−0.0540190
$851$		−	22.5378i		−	0.772587i
$852$		0			0
$853$		−	40.1164i		−	1.37356i	−0.726866	−	0.686779i	$-0.759024\pi$
$853$		−	40.1164i		−	1.37356i	0.726866	−	0.686779i	$-0.240976\pi$
$854$	11.1498			0.381539
$855$		0			0
$856$			25.3475i			0.866361i
$857$	−15.2389			−0.520552			−0.260276	−	0.965534i	$-0.583814\pi$
$857$	−15.2389			−0.520552			−0.260276	+	0.965534i	$0.583814\pi$
$858$		0			0
$859$		−	10.7770i		−	0.367706i	−0.982954	−	0.183853i	$-0.941143\pi$
$859$		−	10.7770i		−	0.367706i	0.982954	−	0.183853i	$-0.0588571\pi$
$860$		−	25.9640i		−	0.885367i
$861$		0			0
$862$			8.08545i			0.275392i
$863$	32.1004			1.09271			0.546355	−	0.837554i	$-0.316015\pi$
$863$	32.1004			1.09271			0.546355	+	0.837554i	$0.316015\pi$
$864$		0			0
$865$	7.79310			0.264973
$866$	92.1932			3.13285
$867$		0			0
$868$			3.48265i			0.118209i
$869$	11.5160			0.390654
$870$		0			0
$871$	−1.01310			−0.0343276
$872$		−	147.674i		−	5.00087i
$873$		0			0
$874$	−89.2615			−3.01932
$875$	−4.21819			−0.142601
$876$		0			0
$877$	28.2520			0.954004			0.477002	−	0.878902i	$-0.341723\pi$
$877$	28.2520			0.954004			0.477002	+	0.878902i	$0.341723\pi$
$878$		−	80.1048i		−	2.70341i
$879$		0			0
$880$		−	29.6091i		−	0.998123i
$881$			8.59043i			0.289419i	0.989474	+	0.144710i	$0.0462248\pi$
$881$			8.59043i			0.289419i	−0.989474	+	0.144710i	$0.953775\pi$
$882$		0			0
$883$	−34.7913			−1.17082			−0.585410	−	0.810737i	$-0.699066\pi$
$883$	−34.7913			−1.17082			−0.585410	+	0.810737i	$0.699066\pi$
$884$		−	1.96770i		−	0.0661808i
$885$		0			0
$886$	−75.1135			−2.52349
$887$			40.3558i			1.35501i	0.735516	+	0.677507i	$0.236940\pi$
$887$			40.3558i			1.35501i	−0.735516	+	0.677507i	$0.763060\pi$
$888$		0			0
$889$			46.7942i			1.56943i
$890$	−19.5862			−0.656531
$891$		0			0
$892$	−70.8364			−2.37178
$893$	13.7931			0.461568
$894$		0			0
$895$	−4.43637			−0.148292
$896$			85.9528i			2.87148i
$897$		0			0
$898$	−11.4934			−0.383541
$899$	−0.210531	−	0.825627i	−0.00702160	−	0.0275362i
$900$		0			0
$901$			7.66723i			0.255432i
$902$		−	60.7071i		−	2.02132i
$903$		0			0
$904$	116.630			3.87904
$905$	−13.5160			−0.449288
$906$		0			0
$907$			8.59463i			0.285380i	0.989767	+	0.142690i	$0.0455752\pi$
$907$			8.59463i			0.285380i	−0.989767	+	0.142690i	$0.954425\pi$
$908$	36.5368			1.21252
$909$		0			0
$910$		−	7.29014i		−	0.241666i
$911$			12.9332i			0.428497i	0.976779	+	0.214249i	$0.0687303\pi$
$911$			12.9332i			0.428497i	−0.976779	+	0.214249i	$0.931270\pi$
$912$		0			0
$913$			19.0207i			0.629492i
$914$			27.8505i			0.921214i
$915$		0			0
$916$		−	126.043i		−	4.16458i
$917$			30.0838i			0.993454i
$918$		0			0
$919$	52.1153			1.71913			0.859563	−	0.511030i	$-0.170736\pi$
$919$	52.1153			1.71913			0.859563	+	0.511030i	$0.170736\pi$
$920$		−	48.2018i		−	1.58917i
$921$		0			0
$922$	35.1724			1.15834
$923$	−6.01894			−0.198116
$924$		0			0
$925$		−	4.04272i		−	0.132924i
$926$			89.8882i			2.95391i
$927$		0			0
$928$	−22.7247	−	89.1184i	−0.745976	−	2.92545i
$929$	26.5993			0.872695			0.436347	−	0.899778i	$-0.356272\pi$
$929$	26.5993			0.872695			0.436347	+	0.899778i	$0.356272\pi$
$930$		0			0
$931$		−	64.3218i		−	2.10806i
$932$	26.8727			0.880246
$933$		0			0
$934$	87.5499			2.86472
$935$	−1.35673			−0.0443697
$936$		0			0
$937$	−14.5291			−0.474646			−0.237323	−	0.971431i	$-0.576270\pi$
$937$	−14.5291			−0.474646			−0.237323	+	0.971431i	$0.576270\pi$
$938$		−	17.8483i		−	0.582767i
$939$		0			0
$940$			12.0773i			0.393918i
$941$	−3.14619			−0.102563			−0.0512815	−	0.998684i	$-0.516331\pi$
$941$	−3.14619			−0.102563			−0.0512815	+	0.998684i	$0.516331\pi$
$942$		0			0
$943$		−	54.4269i		−	1.77238i
$944$	33.8157			1.10061
$945$		0			0
$946$		−	30.9397i		−	1.00594i
$947$		−	51.7960i		−	1.68314i	−0.540145	−	0.841572i	$-0.681631\pi$
$947$		−	51.7960i		−	1.68314i	0.540145	−	0.841572i	$-0.318369\pi$
$948$		0			0
$949$		−	5.90309i		−	0.191622i
$950$	−16.0113			−0.519475
$951$		0			0
$952$	21.3793			0.692907
$953$	10.5542			0.341883			0.170941	−	0.985281i	$-0.445319\pi$
$953$	10.5542			0.341883			0.170941	+	0.985281i	$0.445319\pi$
$954$		0			0
$955$			11.5723i			0.374471i
$956$	76.4113			2.47132
$957$		0			0
$958$	98.1230			3.17021
$959$		−	62.3844i		−	2.01450i
$960$		0			0
$961$	30.9750			0.999192
$962$	6.98690			0.225267
$963$		0			0
$964$	−10.4364			−0.336133
$965$		−	9.84404i		−	0.316891i
$966$		0			0
$967$		−	21.0448i		−	0.676755i	−0.941010	−	0.338378i	$-0.890122\pi$
$967$		−	21.0448i		−	0.676755i	0.941010	−	0.338378i	$-0.109878\pi$
$968$			48.7929i			1.56826i
$969$		0			0
$970$	8.21819			0.263870
$971$			1.01417i			0.0325462i	0.999868	+	0.0162731i	$0.00518012\pi$
$971$			1.01417i			0.0325462i	−0.999868	+	0.0162731i	$0.994820\pi$
$972$		0			0
$973$	15.0320			0.481905
$974$		−	69.6952i		−	2.23318i
$975$		0			0
$976$			12.5865i			0.402883i
$977$	−9.95239			−0.318405			−0.159203	−	0.987246i	$-0.550892\pi$
$977$	−9.95239			−0.318405			−0.159203	+	0.987246i	$0.550892\pi$
$978$		0			0
$979$	−16.8727			−0.539255
$980$	56.3204			1.79909
$981$		0			0
$982$	−70.7699			−2.25836
$983$		−	15.4059i		−	0.491372i	−0.969349	−	0.245686i	$-0.920987\pi$
$983$		−	15.4059i		−	0.491372i	0.969349	−	0.245686i	$-0.0790133\pi$
$984$		0			0
$985$	−5.14982			−0.164087
$986$	−8.21819	+	2.09560i	−0.261720	+	0.0667374i
$987$		0			0
$988$		−	20.0045i		−	0.636428i
$989$		−	27.7390i		−	0.882049i
$990$		0			0
$991$	−8.92035			−0.283364			−0.141682	−	0.989912i	$-0.545251\pi$
$991$	−8.92035			−0.283364			−0.141682	+	0.989912i	$0.545251\pi$
$992$	−2.70217			−0.0857938
$993$		0			0
$994$		−	106.039i		−	3.36334i
$995$	17.7931			0.564079
$996$		0			0
$997$			1.19296i			0.0377814i	0.999822	+	0.0188907i	$0.00601345\pi$
$997$			1.19296i			0.0377814i	−0.999822	+	0.0188907i	$0.993987\pi$
$998$			56.4552i			1.78706i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.b.811.1		6
3.2	odd	2		145.2.c.b.86.6	yes	6
12.11	even	2		2320.2.g.i.1681.2		6
15.2	even	4		725.2.d.c.724.2		12
15.8	even	4		725.2.d.c.724.11		12
15.14	odd	2		725.2.c.e.376.1		6
29.28	even	2	inner	1305.2.d.b.811.6		6
87.17	even	4		4205.2.a.m.1.6		6
87.41	even	4		4205.2.a.m.1.1		6
87.86	odd	2		145.2.c.b.86.1	✓	6
348.347	even	2		2320.2.g.i.1681.5		6
435.173	even	4		725.2.d.c.724.1		12
435.347	even	4		725.2.d.c.724.12		12
435.434	odd	2		725.2.c.e.376.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.c.b.86.1	✓	6	87.86	odd	2
145.2.c.b.86.6	yes	6	3.2	odd	2
725.2.c.e.376.1		6	15.14	odd	2
725.2.c.e.376.6		6	435.434	odd	2
725.2.d.c.724.1		12	435.173	even	4
725.2.d.c.724.2		12	15.2	even	4
725.2.d.c.724.11		12	15.8	even	4
725.2.d.c.724.12		12	435.347	even	4
1305.2.d.b.811.1		6	1.1	even	1	trivial
1305.2.d.b.811.6		6	29.28	even	2	inner
2320.2.g.i.1681.2		6	12.11	even	2
2320.2.g.i.1681.5		6	348.347	even	2
4205.2.a.m.1.1		6	87.41	even	4
4205.2.a.m.1.6		6	87.17	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.68667i q^{2} -5.21819 q^{4} -1.00000 q^{5} +4.21819 q^{7} +8.64620i q^{8} +O(q^{10})\)	\(q-2.68667i q^{2} -5.21819 q^{4} -1.00000 q^{5} +4.21819 q^{7} +8.64620i q^{8} +2.68667i q^{10} +2.31446i q^{11} -0.643274 q^{13} -11.3329i q^{14} +12.7931 q^{16} -0.586195i q^{17} -5.95953i q^{19} +5.21819 q^{20} +6.21819 q^{22} +5.57491 q^{23} +1.00000 q^{25} +1.72826i q^{26} -22.0113 q^{28} +(5.21819 - 1.33061i) q^{29} -0.158221i q^{31} -17.0784i q^{32} -1.57491 q^{34} -4.21819 q^{35} -4.04272i q^{37} -16.0113 q^{38} -8.64620i q^{40} -9.76282i q^{41} -4.97568i q^{43} -12.0773i q^{44} -14.9779i q^{46} +2.31446i q^{47} +10.7931 q^{49} -2.68667i q^{50} +3.35673 q^{52} -13.0796 q^{53} -2.31446i q^{55} +36.4713i q^{56} +(-3.57491 - 14.0195i) q^{58} +2.64327 q^{59} +0.983848i q^{61} -0.425088 q^{62} -20.2978 q^{64} +0.643274 q^{65} +1.57491 q^{67} +3.05888i q^{68} +11.3329i q^{70} +9.35673 q^{71} +9.17663i q^{73} -10.8615 q^{74} +31.0980i q^{76} +9.76282i q^{77} -4.97568i q^{79} -12.7931 q^{80} -26.2295 q^{82} +8.21819 q^{83} +0.586195i q^{85} -13.3680 q^{86} -20.0113 q^{88} +7.29014i q^{89} -2.71345 q^{91} -29.0909 q^{92} +6.21819 q^{94} +5.95953i q^{95} -3.05888i q^{97} -28.9975i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 6 q^{5}+O(q^{10}) \) 6 * q - 6 * q^4 - 6 * q^5	\( 6 q - 6 q^{4} - 6 q^{5} - 4 q^{13} + 26 q^{16} + 6 q^{20} + 12 q^{22} + 8 q^{23} + 6 q^{25} - 56 q^{28} + 6 q^{29} + 16 q^{34} - 20 q^{38} + 14 q^{49} + 20 q^{52} - 28 q^{53} + 4 q^{58} + 16 q^{59} - 28 q^{62} - 46 q^{64} + 4 q^{65} - 16 q^{67} + 56 q^{71} - 40 q^{74} - 26 q^{80} - 56 q^{82} + 24 q^{83} - 4 q^{86} - 44 q^{88} - 16 q^{91} - 48 q^{92} + 12 q^{94}+O(q^{100}) \) 6 * q - 6 * q^4 - 6 * q^5 - 4 * q^13 + 26 * q^16 + 6 * q^20 + 12 * q^22 + 8 * q^23 + 6 * q^25 - 56 * q^28 + 6 * q^29 + 16 * q^34 - 20 * q^38 + 14 * q^49 + 20 * q^52 - 28 * q^53 + 4 * q^58 + 16 * q^59 - 28 * q^62 - 46 * q^64 + 4 * q^65 - 16 * q^67 + 56 * q^71 - 40 * q^74 - 26 * q^80 - 56 * q^82 + 24 * q^83 - 4 * q^86 - 44 * q^88 - 16 * q^91 - 48 * q^92 + 12 * q^94

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